## real analysis – via quotient space and linear map

I'm reading a book called Functional Analysis written by Peter D Lax

Definition M is a linear map of$$X rightarrow U$$.$$N_M$$ is the set of points that are mapped to zero.
$$R_M$$ is the range of M, is the image of X under M in U

He wants me to prove that M maps the quotient space $$X / N_M$$ one on one $$R_M$$

Here is my confusion first, I think the element of the quotient space is like $$y_1, y_2$$ ,so that $$y_1-y_2 in N_M$$So M (y_1-y_2) = 0, then M (y_1) = M (y_n). But that contradicts one on one. Where is my mistake? I do not know how to make mistakes

## Abstract Algebra – Herbrand Quotient

I'm trying to solve exercises from Lang's algebra and stick to a problem with Herbert quotients.

To let $$G$$ Let be a finite cyclic group of orders $$n$$ generated by an element $$sigma$$, Accept that $$G$$ works in an abelian group $$A$$, and let $$f, g: A rightarrow A$$ are the endomorphisms of $$A$$ given by $$f (x) = sigma x – x$$ and $$g (x) = x + sigma x + ldots + sigma ^ {n -1} x$$,

Define the Herbrand quotient by the expression $$q (A) = (A_f: A ^ g) / (A_g: A ^ f)$$assuming both indexes are finite. Now let us assume that $$B$$ is a subgroup of $$A$$ so that $$G B subseteq B$$,

(a) Naturally define an operation of $$G$$ on $$A / B$$,

(b) Prove that $$q (A) = q (B) q (A / B)$$
in the sense that if two of these quotients are finite, then the third and the specified equality holds.

(c) If $$A$$ is finally, show that $$q (A) = 1$$,

The $$A_f, A ^ f$$ in question is the image and the core of $$f$$ as a map of $$A$$ to $$A$$, The same applies to $$B_f$$ and $$B g$$,

I'm stuck right now $$(b)$$ and I do not understand why $$f, g$$ are endomorphisms of groups at the beginning. If $$G$$ affects $$A$$It is not necessary for everyone $$g in G$$ induces an automorphism $$A$$ Law? Thank you in advance!

## Riemannian geometry – cylindrical coordinates in the quotient of the symmetrical space

I am interested in the following situation. Accept $$G / K$$ is a compact and non-symmetrical space $$alpha$$ is the axis of a hyperbolic isometry. I'm interested in the calculation of Hessian's function $$z mapsto d ^ 2 ( alpha, z)$$,

Here is an example of this situation.

To let $$(C, sigma)$$ be the quotient $$mathbb {H} / langle z mapsto lambda ^ 2 z rangle$$ from where $$mathbb {H}$$ is the upper half-plane, which is equipped with its hyperbolic metric $$dz ^ 2 / | In the z | ^ 2$$, To let $$gamma$$ denote the geodesic nucleus. If we take the region in between $$| z | = 1$$ and $$| z | = | lambda | ^ 2$$ be a fundamental domain for $$C$$ in the $$mathbb {H}$$We have orthonormal coordinates $${ xi_1, xi_2 } = {( sin theta) ^ {- 1} partial / partial theta, (r sin theta) ^ {- 1} partial / partial r }$$ You can then count $$textrm {Hess} d ^ 2 ( gamma, cdot) = 2d xi_1 otimes d xi_1 + 2d ( gamma, cdot) tanh d ( gamma, cdot) d xi_2 otimes d xi_2$$ When lifting on the universal lid we get the desired Hessian.

I am looking for something similar. The missing ingredient is a nice set of orthonormal coordinates $$langle alpha rangle backslash G / K$$, Is there a source from which such coordinates are constructed?

## real analysis – Is \$ π: mathcal {C} ^ (M, N) → mathcal {C} ^ (S, N) \$, \$ π (f) = f | _S \$ a quotient map in the \$ mathcal {C} ^ 1 \$ topology?

This question has previously been asked on MSE.

To let $$M, N$$ be smoothly connected distributor (without limitation), wherein $$M$$ is a compact manifold, so we can put a topology in the room $$mathcal C ^ infty (M, N)$$ With $$mathcal {C} ^ 1$$ Whitney topology.

Now think about it $$S subset M$$ a compact sub-variety of $$M$$ with such a limit, that $$text {dim} S = text {dim} M$$Using the same procedure, we can insert a topology $$mathcal C ^ infty (S, N)$$ Use of $$mathcal {C} ^ 1$$ Whitney topology. There is a natural continuous projection of $$mathcal C ^ infty (M, N)$$ on $$mathcal C ^ infty (S, N)$$, determined by

begin {align *} pi: mathcal C ^ infty (M, N) & to mathcal C ^ infty (S, N) \ f & mapsto left.f right | _ {S}. end {align *}

My question: is $$pi$$ an open card or at least a quotient card?

$$mathcal {C} ^ 1$$Whitney topology is also called $$mathcal {C} ^ 1$$Topology.

As noted for the user Adam Chalumeau, in the book "Morris W. Hirsh Differential Topology" the following exercise exists

[Exercise 16, page 41]: To let $$M, N$$ His $$mathcal {C} ^ r$$ Distribution. To let $$V⊂M$$ Then be an open sentence

• The restriction map $$δ: mathcal {C} r (M, N) → mathcal {C} r (V, N)$$ $$δ (f) = f | V$$ is steady for the weak topology, but not always for the strong one.

• $$δ$$ is open for the strong topologies, but not always for the weak ".

Since ours $$M$$ is compact weak topology = strong topology. However, I do not know how to solve this problem, let alone to adapt such proof to the case I want.

## ag.algebraic geometry – Maximum non-branched quotient of \$ E[p]\$ for the action of \$ G _ { mathbb {Q} _p} \$

To let $$E$$ be an elliptic curve with good and ordinary reduction for an odd prime $$p$$,
Accept $$E[p]$$ refers to the $$p$$Fulcrums of $$E$$ and $$G _ { mathbb {Q} _p}: = text {Gal} ( overline { mathbb {Q} _p} / mathbb {Q} _p)$$,

In the article "Selmer Group and Congruences (page 6)" Greenberg says that one can characterize $$widetilde {E}[p]$$ as maximum unbranched quotient of $$E[p]$$ for the action of $$G _ { mathbb {Q} _p}$$ from where $$widetilde {E}$$ denotes the reduction of $$E$$ in the $$mathbb {F} _p$$,

That's because $$p$$ is assumed to be odd and therefore the effect of the inertial subgroup of $$G _ { mathbb {Q} _p}$$ on the kernel of the reduction map $$pi: E[p] longrightarrow widetilde {E}[p]$$ is not trivial.

It is very helpful if someone can explain how$$p$$ ung plays a role to prove the non-trivial effect of the inertial subgroup on the core of the reduction map $$pi$$ ?

## linear algebra – Prove that the Rayleigh quotient of the self-adjoint positive-semidefinite operator with continuous inverse has a positive lower bound.

To let $$V$$ be a Hilbert room. To let $$A: v in V right arrow Av in V$$ Let be a self-adapting continuous bijective linear operator with continuous inverse such that
$$begin {equation} (Av, v) geq 0, quad forall v in V. end {equation}$$
How can you prove that there is a constant? $$alpha> 0$$ so that $$(Av, v) geq alpha || v || ^ 2$$? Can someone help me?

## dg.differential geometry – Kronheimer's results on ALE spaces as Hyperkahler quotient

Background: Kronheimer proved in his two works from the late 80s that each 4-dimensional ALE space is given for example by a Hyperkahler quotient $$X _ {{ zeta_ mathbb {R}}, { zeta_ mathbb {C}}} (Q)$$ where Q is a Dynkin diagram of the type ADE and $${ zeta_ mathbb {R}}, { zeta_ mathbb {C}}$$ are parameters of the Hyperkahler Momentenkarte, which fulfill a certain generic condition. He also proves that if one fixes a graph Q, all of these spaces are diffeomorphic to a space, we call it $$X (Q),$$ what is given as the minimum resolution of $$pi: X (Q) rightarrow mathbb {C} ^ 2 / G,$$ (where G is a particular subset of $$SU (2)$$). From earlier works by Du Val we know the topology of $$X_Q$$ – Its deformation angle is the extraordinary divider $$pi ^ {- 1} (0) = cup_ {i in Q ^ 0} mathbb {C} P ^ 1_i$$ This is a Dynkin Q tree of spheres that intersect transversely according to the graph and whose self-intersections are -2. Particularly, $$H_2 (X (Q)) = oplus_ {i in Q ^ 0} [mathbb{C}P^1_i],$$

Now the question: It seemed to me that Kronheimer also proves the following: in the ALE room $$X _ {{ xi_ mathbb {R}}, 0} (Q)$$ the $$omega_I$$Volumes of these exceptional areas are indicated
$$begin {equation} tag {1} langle omega_I, [mathbb{C}P^1_i] rangle = zeta_ mathbb {R} ^ i end {equation}$$
exactly through the components of the real moment map parameters $$zeta_ mathbb {R} = ( zeta_ { mathbb {R}} ^ i) _ {i in {Q ^ 0}}.$$ Now this seems to be wrong, as are these exceptional areas $$omega_I$$-symplektisch, therefore theirs $$omega_I$$Volumes are positive while moment parameters $$zeta _ { mathbb {R}} ^ i$$ can be negative.

Does anyone know the cure of formula (1) to make it come true?

## \$ F_2 \$ as a quotient of a linear group

I want to find a linear group $$mathbb {Z}$$. $$G$$, so that

$$1 longrightarrow mathbb {Z} ^ 2 longrightarrow G longrightarrow F_2 longrightarrow 1$$ is exactly.

What I'm thinking about is: I know the matrices $$begin {pmatrix} 1 & 2 \ 0 & 1 end {pmatrix}$$. $$begin {pmatrix} 1 & 0 \ 2 & 1 end {pmatrix}$$ Create a free subgroup in SL$$(2, mathbb {Z})$$, also $$mathbb {Z} ^ 2$$ is linear over $$mathbb {Z}$$ and with it the product $$mathbb {Z} ^ 2 times F_2$$ is linear over $$mathbb {Z}$$ also. Is that enough? Is there any meaning for the multiplication $$mathbb {Z}$$ twice, or is there a more canonical example?

## Group theory – quotient of \$ S ^ 1 \$ by dense subgroups

I try to calculate the quotients of the circle group $$S ^ 1$$ through its various subgroups. So far, I have made a rough classification of the subgroups, and all finite subgroups of order $$n$$ are the $$n$$Roots of unity $$Gamma_n$$and all other subgroups are tight in $$S ^ 1$$,

In the case of finite subgroups, the group homomorphism is sent $$a mapsto a ^ n$$ gives an isomorphism between $$S ^ 1 / Gamma_n$$ and $$S ^ 1$$ after the first isomorphism phrase.

I am a bit lost when it comes to a dense subgroup $$H$$; Is it even possible to determine the quotient? $$S ^ 1 / H$$ only know that $$H$$ is a dense subgroup, or would one need a more specific subgroup classification?
How could I proceed to better classify the subgroups or calculate the quotient?

## ag.algebraic geometry – motivational quotient of algebraic diversity

To let $$X$$ to be a variety with one $$G$$Action of an algebraic group on it.

My question refers to a motivating example

https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf

Here is the relevant excerpt:

Here the author discusses an example of $$X / G$$ to explain that it is necessary to shape $$X / G$$ as categorical quotient and not the topologically on.

We consider a motivational example introduced on page 27:

Here we take $$X: = mathbb {C} ^ 2$$ with action of $$G: = mathbb {C} ^ x$$ by multiplication $$lambda cdot (x, y) mapsto ( lambda x, lambda y)$$,

Obviously, the "naive" topological quotient theoretically consists of the lines $${( lambda x, lambda y) vert lambda in mathbb {C} ^ x }$$ and the origin $${(0,0) }$$,

Topologically, the origin lies in the closure of each line.

The QUESTION is why that argument already implies that $$Y: = X / G$$ can not have a structure of a species? I do not understand the author's argument.

If we call with $$p: mathbb {C} ^ 2 to Y$$ The canonical projection map and through (continuity?) This card can not separate orbits, which is why this implies $$Y$$ has no structure of a sort as stated in the extract?

What role is particularly played by the fact that we can not separate the lines from the origin (in purely topologically away) It creates an obstacle to form a diverse structure $$X / G$$?

Comment: I know that there are different ways to derive that when we define $$X / G$$ then it can not have a topological structure. The most common argument is to introduce the invariant ring $$R ^ G$$ and explicitly calculate here. But the main problem of this question is
It made me curious that the given reasoning seems a bit more "elemental" in the sense that in this example it does not explicitly work with the concept of the invariant ring $$R ^ G$$,